Document Type

Article

Journal/Book Title/Conference

Compositio Mathematica

Volume

158

Issue

9

Publisher

Cambridge University Press

Publication Date

10-18-2022

First Page

1878

Last Page

1934

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

Abstract

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden-Weinstein-Meyer reduction, Mikami-Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg-Kazhdan construction of Moore-Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian G-space 𝔐𝐺,𝑆 to each pair (G,S), where G is any Lie group and S βŠ† Lie(𝐺)βˆ— is any submanifold satisfying certain non-degeneracy conditions. The spaces 𝔐𝐺,𝑆 satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonioan G-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

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